Editing Connection form (section)

====Curvature====
The curvature 2-form of the Levi-Civita connection is the matrix (Ω<sub>''i''</sub><sup>''j''</sup>) given by
:<math>
\Omega_i{}^j(\mathbf e) = d\omega_i{}^j(\mathbf e)+\sum_k\omega_k{}^j(\mathbf e)\wedge\omega_i{}^k(\mathbf e).
</math>
For simplicity, suppose that the frame '''e''' is [[Holonomic basis|holonomic]], so that {{nowrap|1=''dθ''<sup>''i''</sup> = 0}}.<ref>In a non-holonomic frame, the expression of curvature is further complicated by the fact that the derivatives dθ<sup>i</sup> must be taken into account.</ref>  Then, employing now the [[summation convention]] on repeated indices,
:<math>\begin{array}{ll}
\Omega_i{}^j &= d(\Gamma^j{}_{qi}\theta^q) + (\Gamma^j{}_{pk}\theta^p)\wedge(\Gamma^k{}_{qi}\theta^q)\\
&\\
&=\theta^p\wedge\theta^q\left(\partial_p\Gamma^j{}_{qi}+\Gamma^j{}_{pk}\Gamma^k{}_{qi})\right)\\
&\\
&=\tfrac12\theta^p\wedge\theta^q R_{pqi}{}^j
\end{array}
</math>
where ''R'' is the [[Riemann curvature tensor]].